1

Bending Stiffness of paper and corrugated board: the connection to caliper and tensile stiffness

Generally, the resistance of objects to a mechanical action is called stiffness. We have already

encountered tensile stiffness as the reaction or resistance of paper to tension. Similarly, the resistance

of sheets or boards to bending action is bending stiffness. Bending stiffness ties in the tensile stiffness

and the caliper of the board or sheet and can in principle, be calculated for tensile test measurements

and caliper. Bending stiffness of paper is important to provide the paper rigidity required for printed

tickets, business cards, folding cartons and in converting processes such as printing or folding.

Conversely, low bending stiffness is desired for towels, tissues and non-wovens where conformability or

drape of the sheet products are desired features. Bending stiffness in corrugated boards limits the

outward panel bulging of boxes under load and contributes to the compression strength of boxes.

Simple elasticity theory provides the formula for the bending stiffness Sb of a one-dimensional beam

which we can apply to the sheet of paper per unit width:

𝑆𝑏 = 𝐸𝑡 × 𝑡2

12

where Et is the elastic modulus times the effective caliper, the tensile modulus from the tensile test.

The effective caliper removes the surface roughness contribution to roughness by using caliper-

measuring platens in the standard gauge covered with soft neoprene. Measurement of paper bending

stiffness is made by applying a moment, the simplest example is the 2-point application where a length

of sheet is fixed at one end and a force is applied to the other end. The defection the test specimen

undergoes in reaction to the force is related through elasticity theory. For the 2 point bending of a

beam of length L and width w, standard Euler mechanics elasticity analysis provides the relationship:

2

mm

mmN60m-mN

22

w

LFSb

between the force F subjecting a beam to a circular arc deflection of α degrees. The commonly used

Taber style of instruments measure the force required to achieve a 15 degree of deflection of a 38 x 50.2

mm test specimen. This feature of Taber instruments is sometimes a source for confusion since it

reports the bending resistance in terms of the bending moment M, and not the bending stiffness. The

relationship between moment and bending stiffness Sb.

𝑆𝑏 = 𝑀𝐿/3𝛼w

where L is the span (50 mm), w is the width (38.1 mm), and α is the bending angle in radians (15°). It is

useful to use this formula to convert from Taber moment units M in gr-cm to bending stiffness Sb in mN-

m. This is done below:

𝑆𝑏 (𝑚𝑁 − 𝑚) = (𝑀 × 9.807 𝑥 10−2)(𝑚𝑁 − 𝑚) 𝐿/3𝛼𝑤

= 𝑀 × 9.807 𝑥 10−2 (𝑚𝑁 − 𝑚) 50( 𝑚𝑚)

3 𝜋

180 15 38.1 (𝑚𝑚)

= 0.1639 M (gr-cm)

The formulas assume that beam strains are in the linear elastic region of the stress strain curve for the

material. This requires the assumption that the deflection is small, the moment point curves the beam

around a circular arc, and that the beam is long relative to its thickness such that out of plane shear

strain is negligible. Therefore, consideration must be given when testing samples that limits in sample

size and deflection are taken to ensure linear elasticity.

The Taber and L&W bending resistance measuring instruments are shown in Figure 1. Both

instruments typically use a 38 x 50 mm test specimen prepared using the punch cutter shown in Figure

2. The length of the test specimen is cut in the direction of interest to be measured, either in the MD or

3

CD for machine made papers. The Taber turns the mounted sample subjected to a counterweight until a

15-degree deflection is attained. Different counter -weight arrangements are applied according to a

table to suit the range of bending resistance being measured. Similarly, but simpler in design, the L&W

instrument bends a

Figure 1. The Taber (left) and L&W 2 point bending stiffness testers commonly used in the paper industry to measure the bending of board and paper samples.

4

Figure 2. Both the Taber and the L&W bending measuring instruments use a punch cutter that prepares 50.2 x 38.1 mm test specimens, the length being along the MD or CD of the sheet.

clamped end of the test specimen to a rotation of 5 degrees and measures the resulting force at the

other end by a contacting load cell. Different specimen lengths and deflection angles can be selected for

both instruments depending on the stiffness of the sample.

An example of measuring the bending stiffness of several samples and their relation to tensile stiffness

and soft and hard caliper follows. Seven different samples were tested Table 1 displays the designation,

grade, basis weight, soft caliper, and density for each sample used in this study.

Table 1: List of the various samples used in this bending stiffness study, basis weights, soft caliper, and density. Basis weight is in units of g/m2, Soft Calper is in microns and density is kg/m3.

Sample ID

Description Basis Weight

g/m2

Soft Caliper

microns

Density

kg/m3

5

A 42# Brown Kraft Linerboard 212 277 766

B 26# Neutral Sulfite Semi-Chemical Medium

130 191 679

C Lightweight bleached kraft 75 92 823

D Newsprint 45 57 808

E Lightweight coated 47 44 1057

F Mylar Transparency Film 146 98 1500

G Synthetic Paper 156 97 1610

These samples were used to acquire hard caliper, soft caliper, a “stack” caliper, and the MD and CD

calculated effective thicknesses. The hard and soft calipers were based on TAPPI methods T 411 and T

551, respectively. The hard caliper was measured on an Emveco 200A and the soft caliper was measured

on an Emveco 210-DH. The “stack” caliper was measured by stacking twelve specimens of a sample and

dividing the result by 12. The expectation of T 411 is that the hard caliper result is a

prediction for the thickness of a stack of specimens.

Figure 3. A comparison of hard and soft caliper for a range of sample listed in Table 1.

6

The comparison of hard and soft caliper shows the largest differences occur for samples of a low density

such as A and B which are also rougher surfaces compared to synthetic (sample G ) or coated magazine

journal paper (sample E) .

The effective mechanical equivalent thickness (te) was also calculated to compare This parameter,

proposed by Setterholm, calculates the thickness from the relationship between tensile stiffness (St) and

bending stiffness (Sb) This is derived as:

t

be

b

t

S

St

tES

tES

12

12

3

(1)

where E is the elastic modulus.

The tensile measurements for the samples in Table 1 were made on an Instron 1122 universal test

machine according to TAPPI method T 494 om-96. The tensile measurements provide values for tensile

stiffness by

w

L

dx

dFS t

max

which is the Instron Series IX™ software tensile stiffness slope algorithm. L is the gauge length (178 mm),

w is the width of the sample (25.4 mm), and te is the soft caliper measured and entered separately.

Bending stiffnesses were measured on the L&W instrument using the default 5-degree deflection.

7

Figure 4. Comparison of the calculated effective Setterholm thickness with soft caliper measurements.

Figure 4 shows that the effective thickness as calculated by the stiffness to bending stiffness ratio

matches the measured caliper using the soft platen method. Therefore, the soft platen caliper can also

be used to calculate the bending stiffness once the tensile stiffness is known as will be shown in the

following. If we measure bending stiffness using different parameter or different instruments how can

we know which is right? In Table 1 below we have the bending stiffness of the variety of samples

described measured by three different methods.

Note for example, that for the paper samples A through E, the stiffness at 30 degrees is smaller than at 5

degrees. The angles and spans for the samples were selected to get sufficient instrument sensitivity.

Sample ID 5 deg L&W 30 deg L&W Taber 5 deg L&W 30 deg L&W Taber

A 9.235 8.106 8.932 5.051 4.466 4.800

B 2.278 1.966 2.012 0.891 0.812 0.799

C 0.358 0.352 0.333 0.144 0.141 0.144

D 0.102 0.099 0.101 0.016 0.017 0.017

E 0.065 0.057 0.058 0.031 0.030 0.027

F 0.431 0.469 0.465 0.417 0.437 0.448

G 0.393 0.409 0.398 0.360 0.390 0.384

MD CD

8

The Taber instrument tests are all done with its default 15 degree deflection. We will use calculations

form elasticity theory to check the accuracy of bending stiffness measurements. Bending stiffness

measurement requires the assumption that the deflection is small, the moment point curves the beam

around a circular arc, and that the beam is long relative to its thickness such that out of plane shear

strain is negligible. These assumptions are not necessarily valid in standard bending stiffness

measurements, so the assumption should be considered in validating a measurement. Stress/Strain

tensile test curves show that strains below 0.2% were within the linear elastic region of the curve. For

the two-point method, the 2 point bending strain ε can be estimated as:

l

tee

120

Table 2 shows a comparison of the angles, spans, and equivalent strains used in the L & W

measurements for the seven different samples. By comparison with stress/strain curves, we should

expect the stiffnesses at 5° to be higher than those at 30° since the elastic modulus at non-linear strains

is lower than those at linear strains. Lightweight grades cannot be measured with the L & W instrument

using default settings (50mm span, 5 degree deflection) so large angle deflections are required at the

risk of underestimating the bending stiffness from non-linear strain.

Table 2. Listing of various test specimen bending stiffness measurement parameters: free spans, resulting corresponding strains at 2 deflections and the Taber ranges used for comparison of results.

Sample ID

L & W spans, mm Bending strains ε, %

Taber Range MD CD

5º deflection

30º deflection

A 25 25 0.14 0.87 10-100

B 20 20 0.13 0.75 1-10

C 15 15 0.080 0.48 1-10

D 10 5 0.074 /.15a 0.44 / .89a 1-10

E 10 10 0.058 0.70 1-10

F 15 15 0.085 0.51 1-10

G 15 15 0.084 0.51 1-10

9

a The two numbers here correspond to the MD 10 and CD 5 mm spans respectively.

Measurements of bending stiffness listed in Table 2 were compared to calculated bending stiffness using

the elastic modulus from tensile measurements by

etw

L

dx

dFE

max

223

3112 e

tEe

btE

tES

using the Instron Series IX™ software modulus algorithm with te the soft caliper measured and entered

separately. Averages denoted by brackets were calculated based on 5 or more repeat measurements

and the term in square brackets reflects the propagation of relative errors based on the standard

deviations denoted by σ.

Figure 1 compares the bending stiffness of seven different papers measured by three different

methods L&W 5 degree bend angle, L&W 30 degree bend angle, Taber stiffness (15 degree bend angle)

to the theoretical bending stiffness calculated using the measured elastic modulus.

10

Figure 1: The theoretical bending stiffness plotted against the experimental stiffnesses by three different methods: L&W 5 degrees, L&W 30 degrees and the Taber instrument. The results from all three methods correlate well to theory calculations, including the lightweight grades.

The results shown in Figure 4 indicate that the various measurements are valid within experimental

error when compared to the expected results from calculations. Thus bending stiffness measurements

can be checked using the combination of tensile and soft caliper measurements using the equations

from linear elasticity theory.

Bending stiffness matters for corrugated boxes since the amount of panel outward bulging is desired to

be limited. The term flexural rigidity is often used for corrugated boards although the relationship

between flexural rigidity and bending stiffness involves a Poisson ratio term which is usually not known

nor measured for corrugated board. In the case of corrugated boards, the so-called sandwich beam

theory for bending stiffness applies to a good approximation:

𝑆𝑏 = 𝐸 ∙ 𝑡 ∙ ℎ2

2

where E and t are the modus and caliper of the outside linerboard and h is the caliper of the board.

In this case, hard caliper measurements suffice. The equation indicates that the medium does not

contribute to the bending stiffness other than providing spacing between the linerboard facings

producing the board caliper h. More accurate calculations show that the contribution of the fluted

medium to bending stiffness of the board structure is about 5%. The sandwich beam approximation

is also useful for consideration for increasing the bending stiffness of thicker paperboards where

the greater gains are obtained when the outside facing layers are made to have high tensile

stiffness.

The arrangement for testing the bending stiffness of corrugated boards uses the 4-point method

shown schematically in the diagram below. The load is applied at the ends of the test specimen by

weights designated as P/2. The board is subjected to moments at either end from the weights P/2

11

Figure 5. Schematic of the principle of the four point bending arrangement for mesuring bending stiffness.

at a fixed distance, in this case 0.13m, from the fulcrum points designated as triangles. The

deflection of the board y is related to the board bending stiffnesss D through the relation:

𝐷 = (

𝑃2

) 0.13 𝐿2

8 𝑤 𝑦

L is the length of the board between the pivot points, typically 0.2 m cut along either the MD or the

CD, w is the width typically 0.1 m.

The equipment is shown in Figure 6. The weights for P/2 are supplied in incremental gram units

Figure 6. 4 point bending instrument made by Lorentzen and Wettre.

12

So for typical situation with boards cut to 12 x 4 inch pieces, the deflection y panel read-out in mm,

the convenient form of the formula becomes simply:

𝐷(𝑁 − 𝑚) = 0.0627 (

𝑃2

) 𝑔𝑟𝑎𝑚𝑠

𝑦 (𝑚𝑚)

For C-flute corrugated board, P/2 is 170 for CD and 320 for MD and deflections are about 1mm or

less providing values of 15 N-m for the MD and 6 N-m for the CD.

The four point bending method elimiminates the effect of shear in the mesurement from the

symmetry of the load application and also by the clamping restraint of the board test piecce. In the

alternative common three point method a test piece board is instead supported at either end and is

simply delfected in the middle and load at the middle measured. For corrugated board MD shear

develops mostly as a relative displacement of the linerboard facings when the test pieces become

bent. CD shear for corrugated is considerbaly restricted by the facings adhered to the flute tips at

the glue lines. Shear increases with shorter test pieces, experiments show that test pieces have to

longer than 26 inches in the MD or longer than 10 inches in the CD to match the highr bending

stiffness values obtained with 4 point bending.

An example of the difference in results between 4 point and 3 point bending results are shown in

the Figure 7 below. In this case, a heavyweight single wall C-flute board was cut into lengths that

were supported at 2 points 14 inches apart and deflected about 2mm at the center using the

crosshead of a tensile testing machine, the arrangement shown in Figure 7. The crosshead was

fitted with a load cell that measured the resulting force P caused by deflecting the board. This

provided the 3 point bending stiffness according to the equation:

𝐷 =𝑃 𝐿3

48 𝑤 𝑦

13

Figure 7. 3 point bending arrangement using an Instron UTM.

The same boards were tested in the 4 point equipment. One series of boards were taken directly off

the corrugator and cut to size on a CAD table, the other series of bords were sent through the

converting slitter-scorer which makes the flaps of a box and a third series were samples cut from

from box blanks made by the folder-gluer.

Figure 7. Comparison of 4 point and 3 point bending stiffness for a heavyweight C flute corrugated board sample at three different points in the converting process as indicated. Eror bars represent the 95% confidence interval of repeat measurements for each average value.

14

All converting stages can introduce some degree of crushing of the board. In Figure 7, it is obvious

that the MD bending stiffness is greater than the CD by about a factor of 2 reflecting the tensile

stiffness orientation of the linerboard facings. The 3 point bending stiffness are lower than the

corresponding 4 point values by 30% or more. Boards that have been subjected to the converting

processes also show lower significant MD values than the CAD cut board. This can be attributed the

effects of loss of shear stiffness caused by crushing of the board which can be also detected by a

correponding change in caliper measurement as shown below:

Figure 8. Comparison of caliper of singe wall board samples at different converting stages.

The same manufacturer was intersted in the possibility of replacing heavyweight single wall board

with lightweight double wall board. The idea here is that the loss in compression strength from

reduction in basis weight can be compensated by an increase in bending stiffness as occurs with the

increase in caliper of double wall board. A description of the composition of the sample set follows

15

Table 3. Decription and composition of a test corrugated board sample set.

The 4 point bending stiffness and the 3 point bending stiffnesses were compared for this sample

set. The 4 point always measures higher than the 3 point, particulary in the MD as Figure 9 shows

below. Some of the lighter linerboards as used in preparing samples B, C, were made using novel

headbox technology producing a sheet with no orientation, this results in MD and CD bending

stiffnesses not being significantly different from each other as is usually the case.

Figure 9. Comparison of 3 point and 4 point bending stiffness measurements for the samle set of corrugated boards.

Sample ID

liner medium liner medium liner total wt.

A - heavy weight single wall 337 112 337 ─ ─ 786

B - super lighweight double wall 88 112 88 112 88 488

C - lightweight double wall 98 112 98 112 98 518

D - lightweight BB but with kraft 98 112 98 112 98 518

E - medium weight double wall 127 112 127 112 127 605

F - heavy weight double wall 151 112 127 112 151 653

G - heavy weight BB "X"-flute 151 112 49 112 151 575

H - high weight single wall kraft 274 112 274 ─ ─ 660

I - mid weight C flute with kraft 254 112 254 ─ ─ 620

J - medium weight kraft B-B 161 112 161 112 161 707

basis weight of board components (g/m2)

16

Bending stiffness of corrugated board and loss of shear stiffness from crushing is of consequnce in

predicitng the stacking strength BCT of vertically loaded boxes. The McKee equation is based on the

analysis of the bending and failure of panels and results in the semi-emprical formula in metric

form:

𝐵𝐶𝑇 (𝑘𝑁) = 0.375 (𝐸𝐶𝑇)0.75 (√𝐷𝑀𝐷𝐷𝐶𝐷).25𝑍.5

with the board edge compression strength ECT in kN/m, the MD and bending stiffnesses of the

board DMD and DCD in N-m, and the box perimeter ( 2 x length+width) Z in meters. The geometric

mean of the bending stiffness term √𝐷𝑀𝐷𝐷𝐶𝐷 , is raised to the ¼ , power so that a 10% change in

bending stiffness will affect BCT only to about 2.5 %. The change in the geometric mean bending

stiffness for the set of boxes averages about 18% smaller between 3 point and 4 point

measurements, thus BCT predicted by the formula can be expected to be about 5% smaller when

using 3 point bending stiffness values.

Figure 10. Comparison of actual BCT with the BCT predicted by the McKee formula using 3 point or 4 point bending stifness values.

17

Although all BCT measurements and predictions are withing experimental agreement, Figure 10

shows that when using 3 point bending stiffness the McKee formula does become a more accurate

predictor of the actual BCT values. The mean difference between predicted and actual BCT values is

367 N when using 4 point values compared to a difference of 181 N when 3point bending stiffness

is used instead. Since boxes subjected under vertical load are not constrained to restrict MD shear,

three point bending appears to be more relevant towards a more acurate prediction of BCT using

the McKee formula.

The question remains what should be the optimal 3 point bending stiffness test span to be

relevant as a predictor for a particular box. For long boxes shear stiffness should not matter since 3

point bending stiffness is about the same as true 4 point stiffness once the length exceeds 60cm in

the MD. In fact a formula to convert from 3 point D3 measured at 2 different lengths L1 and L2 to 4

point D4 , L1 > L2 is:

𝐷4 = 𝐷3,1𝐷3,2 (𝐿1

2− 𝐿22)

𝐷3,2𝐿12 − 𝐷3,1 𝐿2

2

For the case visited in Figures 7, 9 and 10 the box dimensions were 61 x 41 x 66 cm so that MD

shear should not be significant however the selected test length of 36 cm still provides a better BCT

prediction.

Deflection in bending stifness measurements shouldbe limited to allow the assumption of linear

elasticity theory to apply. In particular, the strain ε of the outerliner must be kept below 0.05%

which can be calculated using the formula:

𝜀(%) =400 ∙ 𝑦(𝑚𝑚) ∙ 𝑡(𝑚𝑚)

𝐿2

18

where the strain is in %, the deflection y is in mm, t is the board caliper and L is the test span

between the 2 fulcrum points.

Summary

2 point bending stiffness is used for paper and some board materials. 4 point bending stiffness is

used for corrugated board. Both measurements depend on the tensile stiffness E x t . For paper, it

more accurate to use the soft platen caliper if bending stiffness Et3/12 is to be calculated as a check

of accuracy of the measurements. In this case, the modulus E can be determine from the analysis of

a tensile test and the calculation of D = Et3/12 can be compared to measurements of D. For

corrugated board, the bending stiffness is approximated by the sandwich beam model Eth2/2

where Et here is now the tensile stiffness of the outer liners and h is the board caliper. 3 point

bending provides smaller values than 4 point depending on the test piece length because of the

effect of shear. 3 point bending stifffness values may provide a more accurate prediction of BCT

when used in the McKee equation.

19